\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{10}   &  3.0 & +  6.00 x_{1} & -9.00 x_{2} &   & +  3.00 x_{4} & + 10.00 x_{5} & +  7.00 x_{6} & -9.00 x_{7} & +  1.00 x_{8} & -2.00 x_{9}\\
 x_{11}   &  -1.0 & +  8.00 x_{1} & +  1.00 x_{2} & +  4.00 x_{3} & +  4.00 x_{4} & -9.00 x_{5} & -4.00 x_{6} & -9.00 x_{7} &   & +  4.00 x_{9}\\
 x_{12}   &  1.0 & -6.00 x_{1} & + 10.00 x_{2} & -4.00 x_{3} & -6.00 x_{4} & +  4.00 x_{5} & +  9.00 x_{6} & -8.00 x_{7} & +  6.00 x_{8} & +  1.00 x_{9}\\
 x_{13}   &  -2.0 & +  1.00 x_{1} & -9.00 x_{2} & +  2.00 x_{3} & -5.00 x_{4} & +  2.00 x_{5} & +  8.00 x_{6} & +  9.00 x_{7} & +  9.00 x_{8} & -4.00 x_{9}\\
 x_{14}   &  0.0 & -9.00 x_{1} & -6.00 x_{2} & -10.00 x_{3} & +  3.00 x_{4} & -3.00 x_{5} & -1.00 x_{6} & -6.00 x_{7} & -4.00 x_{8} & + 10.00 x_{9}\\
\hline
z    &  0.0  &   & -4.00 x_{2} & -1.00 x_{3} & -5.00 x_{4} & +  5.00 x_{5} & -3.00 x_{6} & -5.00 x_{7} & +  4.00 x_{8} & +  5.00 x_{9}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{9}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,14$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & -6.00 y_{10} & -8.00 y_{11} & +  6.00 y_{12} & -1.00 y_{13} & +  9.00 y_{14}\\
 y_{2}   &  1.0 & +  9.00 y_{10} & -1.00 y_{11} & -10.00 y_{12} & +  9.00 y_{13} & +  6.00 y_{14}\\
 y_{3}   &  1.0  &   & -4.00 y_{11} & +  4.00 y_{12} & -2.00 y_{13} & + 10.00 y_{14}\\
 y_{4}   &  1.0 & -3.00 y_{10} & -4.00 y_{11} & +  6.00 y_{12} & +  5.00 y_{13} & -3.00 y_{14}\\
 y_{5}   &  1.0 & -10.00 y_{10} & +  9.00 y_{11} & -4.00 y_{12} & -2.00 y_{13} & +  3.00 y_{14}\\
 y_{6}   &  1.0 & -7.00 y_{10} & +  4.00 y_{11} & -9.00 y_{12} & -8.00 y_{13} & +  1.00 y_{14}\\
 y_{7}   &  1.0 & +  9.00 y_{10} & +  9.00 y_{11} & +  8.00 y_{12} & -9.00 y_{13} & +  6.00 y_{14}\\
 y_{8}   &  1.0 & -1.00 y_{10} &   & -6.00 y_{12} & -9.00 y_{13} & +  4.00 y_{14}\\
 y_{9}   &  1.0 & +  2.00 y_{10} & -4.00 y_{11} & -1.00 y_{12} & +  4.00 y_{13} & -10.00 y_{14}\\
\hline
z    &  -0 & -3.00 y_{10} & +  1.00 y_{11} & -1.00 y_{12} & +  2.00 y_{13} &   \\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 4 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{11}   &  0.196666666667 & -0.35 y_{10} & -0.08 y_{1} & +  0.23 y_{12} & -0.03 y_{8} & -0.09 y_{9}\\
 y_{2}   &  2.60333333333 & + 11.95 y_{10} & +  0.48 y_{1} & -21.83 y_{12} & -1.37 y_{8} & -0.71 y_{9}\\
 y_{3}   &  0.72 & +  4.91 y_{10} & +  0.69 y_{1} & -0.71 y_{12} & -0.03 y_{8} & -0.39 y_{9}\\
 y_{4}   &  0.706666666667 & -2.43 y_{10} & +  0.30 y_{1} & +  2.16 y_{12} & -0.41 y_{8} & +  0.40 y_{9}\\
 y_{5}   &  2.71666666667 & -12.18 y_{10} & -0.66 y_{1} & -1.74 y_{12} & -0.12 y_{8} & -0.94 y_{9}\\
 y_{6}   &  0.693333333333 & -8.44 y_{10} & -0.43 y_{1} & -1.30 y_{12} & +  0.88 y_{8} & -0.14 y_{9}\\
 y_{7}   &  1.93 & +  7.56 y_{10} & -0.66 y_{1} & + 14.99 y_{12} & +  0.66 y_{8} & -0.93 y_{9}\\
 y_{13}   &  0.146666666667 & +  0.05 y_{10} & +  0.02 y_{1} & -0.92 y_{12} & -0.13 y_{8} & -0.04 y_{9}\\
 y_{14}   &  0.08 & +  0.36 y_{10} & +  0.04 y_{1} & -0.56 y_{12} & -0.04 y_{8} & -0.08 y_{9}\\
\hline
z    &  0.49 & -3.25 y_{10} & -0.05 y_{1} & -2.60 y_{12} & -0.29 y_{8} & -0.16 y_{9}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{10}   &  3.25333333333 & +  0.35 x_{11} & -11.95 x_{2} & -4.91 x_{3} & +  2.43 x_{4} & + 12.18 x_{5} & +  8.44 x_{6} & -7.56 x_{7} & -0.05 x_{13} & -0.36 x_{14}\\
 x_{1}   &  0.0466666666667 & +  0.08 x_{11} & -0.48 x_{2} & -0.69 x_{3} & -0.30 x_{4} & +  0.66 x_{5} & +  0.43 x_{6} & +  0.66 x_{7} & -0.02 x_{13} & -0.04 x_{14}\\
 x_{12}   &  2.59666666667 & -0.23 x_{11} & + 21.83 x_{2} & +  0.71 x_{3} & -2.16 x_{4} & +  1.74 x_{5} & +  1.30 x_{6} & -14.99 x_{7} & +  0.92 x_{13} & +  0.56 x_{14}\\
 x_{8}   &  0.286666666667 & +  0.03 x_{11} & +  1.37 x_{2} & +  0.03 x_{3} & +  0.41 x_{4} & +  0.12 x_{5} & -0.88 x_{6} & -0.66 x_{7} & +  0.13 x_{13} & +  0.04 x_{14}\\
 x_{9}   &  0.156666666667 & +  0.09 x_{11} & +  0.71 x_{2} & +  0.39 x_{3} & -0.40 x_{4} & +  0.94 x_{5} & +  0.14 x_{6} & +  0.93 x_{7} & +  0.04 x_{13} & +  0.08 x_{14}\\
\hline
z    &  -0.49 & -0.20 x_{11} & -2.60 x_{2} & -0.72 x_{3} & -0.71 x_{4} & -2.72 x_{5} & -0.69 x_{6} & -1.93 x_{7} & -0.15 x_{13} & -0.08 x_{14}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{10}   &  3.25333333333 & +  0.35 x_{11} & -11.95 x_{2} & -4.91 x_{3} & +  2.43 x_{4} & + 12.18 x_{5} & +  8.44 x_{6} & -7.56 x_{7} & -0.05 x_{13} & -0.36 x_{14}\\
 x_{1}   &  0.0466666666667 & +  0.08 x_{11} & -0.48 x_{2} & -0.69 x_{3} & -0.30 x_{4} & +  0.66 x_{5} & +  0.43 x_{6} & +  0.66 x_{7} & -0.02 x_{13} & -0.04 x_{14}\\
 x_{12}   &  2.59666666667 & -0.23 x_{11} & + 21.83 x_{2} & +  0.71 x_{3} & -2.16 x_{4} & +  1.74 x_{5} & +  1.30 x_{6} & -14.99 x_{7} & +  0.92 x_{13} & +  0.56 x_{14}\\
 x_{8}   &  0.286666666667 & +  0.03 x_{11} & +  1.37 x_{2} & +  0.03 x_{3} & +  0.41 x_{4} & +  0.12 x_{5} & -0.88 x_{6} & -0.66 x_{7} & +  0.13 x_{13} & +  0.04 x_{14}\\
 x_{9}   &  0.156666666667 & +  0.09 x_{11} & +  0.71 x_{2} & +  0.39 x_{3} & -0.40 x_{4} & +  0.94 x_{5} & +  0.14 x_{6} & +  0.93 x_{7} & +  0.04 x_{13} & +  0.08 x_{14}\\
\hline
z    &  1.93 & +  0.54 x_{11} & +  5.06 x_{2} & +  1.04 x_{3} & -5.39 x_{4} & + 10.18 x_{5} & -5.81 x_{6} & -2.99 x_{7} & +  0.69 x_{13} & +  0.56 x_{14}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{10}   &  3.25333333333 & +  0.35 x_{11} & -11.95 x_{2} & -4.91 x_{3} & +  2.43 x_{4} & + 12.18 x_{5} & +  8.44 x_{6} & -7.56 x_{7} & -0.05 x_{13} & -0.36 x_{14}\\
 x_{1}   &  0.0466666666667 & +  0.08 x_{11} & -0.48 x_{2} & -0.69 x_{3} & -0.30 x_{4} & +  0.66 x_{5} & +  0.43 x_{6} & +  0.66 x_{7} & -0.02 x_{13} & -0.04 x_{14}\\
 x_{12}   &  2.59666666667 & -0.23 x_{11} & + 21.83 x_{2} & +  0.71 x_{3} & -2.16 x_{4} & +  1.74 x_{5} & +  1.30 x_{6} & -14.99 x_{7} & +  0.92 x_{13} & +  0.56 x_{14}\\
 x_{8}   &  0.286666666667 & +  0.03 x_{11} & +  1.37 x_{2} & +  0.03 x_{3} & +  0.41 x_{4} & +  0.12 x_{5} & -0.88 x_{6} & -0.66 x_{7} & +  0.13 x_{13} & +  0.04 x_{14}\\
 x_{9}   &  0.156666666667 & +  0.09 x_{11} & +  0.71 x_{2} & +  0.39 x_{3} & -0.40 x_{4} & +  0.94 x_{5} & +  0.14 x_{6} & +  0.93 x_{7} & +  0.04 x_{13} & +  0.08 x_{14}\\
\hline
z    &  1.93 & +  0.54 x_{11} & +  5.06 x_{2} & +  1.04 x_{3} & -5.39 x_{4} & + 10.18 x_{5} & -5.81 x_{6} & -2.99 x_{7} & +  0.69 x_{13} & +  0.56 x_{14}\\
\end{array}\]


 $ x_{2} $ enters and $ x_{1} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{10}   &  2.09677419355 & -1.69 x_{11} & + 24.78 x_{1} & + 12.28 x_{3} & +  9.81 x_{4} & -4.07 x_{5} & -2.25 x_{6} & -23.92 x_{7} & +  0.39 x_{13} & +  0.63 x_{14}\\
 x_{2}   &  0.0967741935484 & +  0.17 x_{11} & -2.07 x_{1} & -1.44 x_{3} & -0.62 x_{4} & +  1.36 x_{5} & +  0.89 x_{6} & +  1.37 x_{7} & -0.04 x_{13} & -0.08 x_{14}\\
 x_{12}   &  4.70967741935 & +  3.49 x_{11} & -45.28 x_{1} & -30.69 x_{3} & -15.65 x_{4} & + 31.42 x_{5} & + 20.82 x_{6} & + 14.89 x_{7} & +  0.11 x_{13} & -1.25 x_{14}\\
 x_{8}   &  0.41935483871 & +  0.26 x_{11} & -2.84 x_{1} & -1.94 x_{3} & -0.44 x_{4} & +  1.99 x_{5} & +  0.35 x_{6} & +  1.22 x_{7} & +  0.08 x_{13} & -0.07 x_{14}\\
 x_{9}   &  0.225806451613 & +  0.21 x_{11} & -1.48 x_{1} & -0.64 x_{3} & -0.85 x_{4} & +  1.91 x_{5} & +  0.78 x_{6} & +  1.91 x_{7} & +  0.01 x_{13} & +  0.02 x_{14}\\
\hline
z    &  2.41935483871 & +  1.41 x_{11} & -10.49 x_{1} & -6.23 x_{3} & -8.51 x_{4} & + 17.06 x_{5} & -1.29 x_{6} & +  3.93 x_{7} & +  0.51 x_{13} & +  0.14 x_{14}\\
\end{array}\]


 $ x_{5} $ enters and $ x_{10} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  0.515288788222 & -0.41 x_{11} & +  6.09 x_{1} & +  3.02 x_{3} & +  2.41 x_{4} & -0.25 x_{10} & -0.55 x_{6} & -5.88 x_{7} & +  0.10 x_{13} & +  0.16 x_{14}\\
 x_{2}   &  0.797281993205 & -0.39 x_{11} & +  6.21 x_{1} & +  2.66 x_{3} & +  2.66 x_{4} & -0.33 x_{10} & +  0.14 x_{6} & -6.62 x_{7} & +  0.09 x_{13} & +  0.13 x_{14}\\
 x_{12}   &  20.9009060023 & -9.54 x_{11} & + 146.10 x_{1} & + 64.11 x_{3} & + 60.11 x_{4} & -7.72 x_{10} & +  3.45 x_{6} & -169.79 x_{7} & +  3.14 x_{13} & +  3.62 x_{14}\\
 x_{8}   &  1.44280860702 & -0.56 x_{11} & +  9.25 x_{1} & +  4.05 x_{3} & +  4.35 x_{4} & -0.49 x_{10} & -0.75 x_{6} & -10.46 x_{7} & +  0.27 x_{13} & +  0.23 x_{14}\\
 x_{9}   &  1.2100792752 & -0.58 x_{11} & + 10.15 x_{1} & +  5.12 x_{3} & +  3.76 x_{4} & -0.47 x_{10} & -0.28 x_{6} & -9.32 x_{7} & +  0.19 x_{13} & +  0.32 x_{14}\\
\hline
z    &  11.2089467724 & -5.66 x_{11} & + 93.40 x_{1} & + 45.23 x_{3} & + 32.62 x_{4} & -4.19 x_{10} & -10.72 x_{6} & -96.33 x_{7} & +  2.15 x_{13} & +  2.79 x_{14}\\
\end{array}\]


 $ x_{1} $ enters and Unbounded Dictionary!


 $ x_{1} $ enters and Unbounded Dictionary!
\end{document}
